Classification of dynamical systems based on a decomposition of their vector fields

We present a method for the global classification of dynamical systems based on a specific decomposition of their vector fields. Every differentiable vector field on Rn can be decomposed uniquely in the sum of 2 systems: one gradient and one that leaves invariant the spheres Sn−1. We show that, under some conditions, the topological class of a vector field is determined by the topological classes of its summands. We illustrate this method by applying it to a number of vector fields, among them being some members of the so-called Lorenz family. The advantage of such a classification is that equivalent flows exhibit qualitatively the same dynamical phenomena as their parameters are varied.